Topological Dynamical Systems Methods in Early-Universe Cosmologies

This dissertation describes some important problems that we have tried to solve with respect to the state of the early universe, that is, the universe shortly after the Big Bang. Standard early-universe cosmological approaches almost always assume a perfect-fluid isotropic and spatially homogeneous Friedmann-Lemaˆıtre- Robertson-Walker (FLRW) model to study the universe’s evolution. The problem is that the early universe was in a hot, dense, and unstable state. Hence, perfect-fluid models which assume no dissipation may not be accurate at such early epochs in the universe’s evolution. Our approach is to introduce terms in the Einstein field equations that allow the representation of these dissipative/viscous effects. In addition, we relax the condition of isotropy to obtain a class of anisotropic and spatially homogeneous cosmological models, known as the Bianchi models. Our research then is largely focused on studying the dynamics of these Bianchi models in the presence of viscous effects. We feel that studying the early universe in this context is more fruitful than the standard approaches mainly because our models are more realistic representations of the conditions of the early universe. Our technique for studying these models is also quite different than the standard approaches in the literature, in that, we use topological dynamical systems theory to study the early and late-time asymptotic behaviour of the cosmological model under consideration. Our work in this regard has been quite successful, and has led to a number of publications in the Physical Review which are listed in the main dissertation document.